Model a torch battery of length l to be made up of a thin cylindrical bar of radius ‘a’ and a concentric thin cylindrical shell of radius ‘b’ filled in between with an electrolyte of resistivity ρ (see$$ $$figure). If the battery is connected to a resistance of value R, the maximum Joule heating in R will take place for :

Question

Model a torch battery of length l to be made up of a thin cylindrical bar of radius ‘a’ and a concentric thin cylindrical shell of radius ‘b’ filled in between with an electrolyte of resistivity ρ  (see$$ $$figure). If the battery is connected to a resistance of value R, the maximum Joule heating in R will take place for :

Infinity Learn · Accepted Answer

Maximum joule heating in R is possible when internal resistance $$=$$ external resistanceConsider a vertical cylindrical strip of thickness dx and radius x.​Its small resistance dR can be given as, $$dR=$$ρ$$dx2$$$$π$$xl​After integrating we get, $$Rinternal=$$∫ρ$$abdx2$$$$π$$$$xl=$$ρ$$2$$$$π$$l $$abdxx=$$ρ$$2$$$$π$$$$llnxab=$$ρ$$2$$$$π$$llnb−$$lna=$$ρ$$2$$$$π$$$$llnba=Rexternal$$

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